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<TITLE>Financial Utility Documentation</TITLE>
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<H1>Financial Transaction Utility</H1>
<!-- <p>Copyright (C) 1990 - 2000 Terry D. Boldt, All Rights Reserved -->
<HR size=4>
<dl>
<dt><A HREF="#FinCalc">Financial Calculator</A></dt>
<dt><A HREF="#TimeValue">Time Value of Money</A></dt>
  <dd><A HREF="#SimpleInterest">Simple Interest</A></dd>
  <dd><A HREF="#CompoundInterest">Compound Interest</A></dd>
  <dd><A HREF="#PeriodicPayments">Periodic Payments</A></dd>
  <dd><A HREF="#FinTrans">Financial Transactions</A></dd>
<dt><A HREF="#StandardFinConv">Standard Financial Conventions</A></dt>
<dt><A HREF="#CashFlowDiag">Cash Flow Diagrams</A></dt>
  <dd><A HREF="#Appreciation">Appreciation</A></dd>
  <dd><A HREF="#Annuity">Annuity</A></dd>
  <dd><A HREF="#Amortization">Amortization</A></dd>
  <dd><A HREF="#Lease">Annuity</A></dd>
<dt><A HREF="#Interest">Interest</A></dt>
  <dd><A HREF="#CompFreq">Compounding Frequency</A></dd>
  <dd><A HREF="#PayFreq">Payment Frequency</A></dd>
  <dd><A HREF="#DiscIntNARtoEIR">NAR to EIR for Discrete Interest Periods</A></dd>
  <dd><A HREF="#ContIntNARtoEIR">NAR to EIR for Continuous Interest</A></dd>
  <dd><A HREF="#NorVal">Normal CF/PF Values</A></dd>
  <dd><A HREF="#DiscIntEIRtoNAR">EIR to NAR for Discrete Interest Periods</A></dd>
  <dd><A HREF="#ContIntEIRtoNAR">EIR to NAR for Continuous Compounding</A></dd>
<dt><A HREF="#FinEquation">Financial Equation</A></dt>
  <dd><A HREF="#FinDeriv">Financial Equation Derivation</A></dd>
<dt><A HREF="#AmortSched">Amortization Schedules</A></dt>
  <dd><A HREF="#ED_IP_Dates">Effective and Initial Payment Dates</A></dd>
  <dd><A HREF="#Eff_PV">Effective Present Value</A></dd>
  <dd><A HREF="#iterative_soltn">Iterative Amortization Schedule</A></dd>
  <dd><A HREF="#AnnualSum">Annual Summary</A></dd>
  <dd><A HREF="#FinalPayment">Final Payment Calculation</A></dd>
  <dd><A HREF="#AmortCases">Amortization Cases</A></dd>
  <dl><dt></dt>
    <dd><A HREF="#ConstOrigData">Constant Repayment to Principal, Original Data</A></dd>
    <dd><A HREF="#ConstNewData">Constant Repayment to Principal, Delayed Repayment</A></dd>
    <dd><A HREF="#OrigData">Original Data Schedule</A></dd>
    <dd><A HREF="#NewFinalPayment">Recomputed Final Payment</A></dd>
    <dd><A HREF="#NewPayment">Recomputed Periodic Payment</A></dd>
    <dd><A HREF="#NewTerm">Recomputed Term</A></dd>
  </dl>
  <dd><A HREF="#DisplaySched">Amortization Schedule Display</A></dd>
<dt><A HREF="#Usage">Financial Calculator Usage</A></dt>
  <dd><A HREF="#FinCommands">Calculator Commands</A></dd>
  <dd><A HREF="#CalcInput">Calculator Input</A></dd>
  <dd><A HREF="#CalcFun">Calculator Functions</A></dd>
  <dd><A HREF="#UserVar">User Defined Variables</A></dd>
  <dd><A HREF="#Rounding">Rounding</A></dd>
<dt><A HREF="#Examples">Examples</A></dt>
  <dd><A HREF="#SimpleInt">Simple Interest </A></dd>
  <dd><A HREF="#CompundInt">Compound Interest</A></dd>
  <dd><A HREF="#PeriodicPmt">Periodic Payment</A></dd>
  <dd><A HREF="#ConvMortg">Conventional Mortgage</A></dd>
  <dd><A HREF="#FinalPmt">Final Payment</A></dd>
  <dd><A HREF="#AS_AnnualSum">Conventional Mortgage Amortization Schedule - Annual Summary</A></dd>
  <dd><A HREF="#AS_PeriodicPmt">Conventional Mortgage Amortization Schedule - Periodic Payment Schedule</A></dd>
  <dd><A HREF="#AS_VarAdvPmt">Conventional Mortgage Amortization Schedule - Variable Advanced Payments</A></dd>
  <dd><A HREF="#AS_ConstAdvPmt">Conventional Mortgage Amortization Schedule - Constant Advanced Payments</A></dd>
  <dd><A HREF="#BalloonPmt">Balloon Payment</A></dd>
  <dd><A HREF="#CanadianMortg">Canadian Mortgage</A></dd>
  <dd><A HREF="#EuropeanMortg">European Mortgage</A></dd>
  <dd><A HREF="#BiWeeklySav">Bi-weekly Savings</A></dd>
  <dd><A HREF="#PV_AnnuiytDue">Present Value - Annuity Due</A></dd>
  <dd><A HREF="#EffRate">Effective Rate - 365/360 Basis</A></dd>
  <dd><A HREF="#MortgPoints">Mortgage with "Points"</A></dd>
  <dd><A HREF="#EquivPmt">Equivalent Payments</A></dd>
  <dd><A HREF="#Perpetuity">Perpetuity - Continuous Compounding</A></dd>
  <dd><A HREF="#DevCo">Investment Return</A></dd>
  <dd><A HREF="#Retiement">Retirement Investment</A></dd>
  <dd><A HREF="#PropVal">Property Values</A></dd>
  <dd><A HREF="#CollegeExpenses">College Expenses</A></dd>
  <dd><A HREF="#EffAPY">Certificate of Deposit, Annual Percentage Yield</A></dd>
<dt><A HREF="#refs">References</A></dt>
</dl>
<HR size=4>
<A NAME="FinCalc">
<H1>Financial Calculator</H1></A>
<P>Financial Calculator
<!-- <p>Copyright (C) 1990 - 2000 Terry D. Boldt, All Rights Reserved -->

<!-- <p>This version for use WITH ANSI.SYS display driver -->

<p>This is  a complete  financial computation  utility to  solve for  the five
standard financial values: n, %i, PV, PMT and FV
<ul>
<ul>
<li>n   == number of payment periods
<li>%i  == nominal interest rate, NAR, charged
<li>PV  == Present Value of money
<li>PMT == Periodic Payment
<li>FV  == Future Value of money
</ul>
</ul>

<p>In addition, four additional parameters may be specified:
<ol>
<li>Compounding Frequency per year, CF. The number of times the interest is compounded
during the year. The default is 12. The compounding frequency per year may be
different from the Payment Frequency per year

<li>Payment Frequency per year, PF. The number of payments made in a year. Default is 12.

<li>Discrete or continuous compounding, disc. The default is discrete compounding.

<li>Payments may be at the beginning or end of the payment period, beg. The default is for
payments to be made at the end of the payment period.
</ol>

<p>When an amortization schedule is desired, the financial transaction Effective Date, ED,
and Initial Payment Date, IP, must also be entered.

<p>Canadian  and  European  style  mortgages  can  be  handled  in  a simple,
straight-forward manner. Standard financial sign conventions are used:

<hr size=4>
<p align="center">"Money paid out is Negative, Money received is Positive"
<hr size=4>

<A NAME="TimeValue">
<H1>Time Value of Money</H1></A>
<p>If you borrow money, you can expect  to pay rent or interest for its  use;
conversely you expect to receive rent interest on money you loan or invest.
When you rent property, equipment,  etc., rental payments are normal;  this
is  also  true  when  renting  or  borrowing  money.  Therefore,  money  is
considered to have a "time value". Money available now, has a greater value
than money available at some future date because of its rental value or the
interest that it can produce during the intervening period.

<A NAME="SimpleInterest">
<H2>Simple Interest</H2></A>
<p>If you loaned $800 to  a friend with an agreement  that at the end of  one
year he would  would repay you  $896, the "time  value" you placed  on your
$800 (principal) was $96 (interest) for  the one year period (term) of  the
loan. This  relationship of  principal, interest,  and time (term) is  most
frequently expressed as an Annual  Percentage Rate (APR). In this  case the
APR  was  12.0%  [(96/800)*100].  This  example  illustrates the four basic
factors involved  in a  simple interest  case. The  time period (one year),
rate (12.0%  APR), present  value of  the principal  ($800) and  the future
value of the principal including interest ($896).

<A NAME="CompoundInterest">
<H2>Compound Interest</H2></A>
<p>In many cases the interest charge is computed periodically during the term
of  the  agreement.  For  example,  money  left  in a savings account earns
interest that  is periodically  added to  the principal  and in  turn earns
additional interest during succeeding periods. The accumulation of interest
during  the  investment  period  represents  compound interest. If the loan
agreement you  made with  your friend  had specified  a "compound  interest
rate" of  12% (compounded  monthly) the  $800 principal  would have  earned
$101.46 interest for the  one year period. The  value of the original  $800
would be increased  by 1% the  first month to  $808 which in  turn would be
increased  by  1%  to  816.08 the second month,  reaching a future value of
$901.46 after the twelfth iteration. The monthly compounding of the nominal
annual rate (NAR) of 12% produces an effective Annual Percentage Rate (APR)
of 12.683% [(101.46/800)*100].  Interest may be  compounded at any  regular
interval; annually, semiannually, monthly, weekly, daily, even continuously
(a specification in some financial models).

<A NAME="PeriodicPayments">
<H2>Periodic Payments</H2></A>
<p>When money is loaned for longer  periods of time, it is customary  for the
agreement to require the borrower  to make periodic payments to  the lender
during the term of the loan. The payments may be only large enough to repay
the interest,  with the  principal due  at the  end of  the loan period (an
interest only loan), or large enough  to fully repay both the interest  and
principal during the term of the loan (a fully amoritized loan). Many loans
fall somewhere between, with payments that do not fully cover repayment  of
both the principal and interst. These loans require a larger final  payment
(balloon)  to  complete  their  amortization.  Payments  may  occur  at the
beginning or end of a payment period. If you and your friend had agreed  on
monthly repayment of  the $800 loan  at 12% NAR  compounded monthly, twelve
payments of $71.08 for a total of $852.96 would be required to amortize the
loan. The $101.46  interest from the  annual plan is  more than the  $52.96
under the monthly plan because under the monthly plan your friend would not
have had the use of $800 for a full year.

<A NAME="FinTrans">
<H2>Financial Transactions</H2></A>
<p>The  above  paragraphs  introduce  the  basic  factors  that  govern  most
financial  transactions;  the  time  period,  interest rate, present value,
payments and  the future  value. In  addition, certain  conventions must be
adhered to: the interest rate must be relative to the compounding frequency
and payment periods, and the term must be expressed as the total number  of
payments (or compounding periods if there are no payments). Loans,  leases,
mortgages, annuities, savings plans, appreciation, and compound growth  are
among the many financial problems that can be defined in these terms. Some
transactions do not involve payments, but  all of the other factors play  a
part in "time value of money" transactions. When any one of the five  (four
- if no payments are involved)  factors is unknown, it can be  derived from
formulas using the known factors.

<A NAME="StandardFinConv">
<h1>Standard Financial Conventions</h1></A>
<p>The Standard Financial Conventions are:

<ul>
<ul>
<li>Money RECEIVED is a  POSITIVE value and is  represented by an arrow  above
the line

<li>Money PAID OUT is  a NEGATIVE value and  is represented by an  arrow below
the line.
</ul>
</ul>

<A NAME="CashFlowDiag">
<h1>Cash Flow Diagrams</h1></A>
<p>If payments are  a part of  the transaction, the  number of payments  must
equal the number of periods (n).

<p>Payments may be  represented as occurring  at the end  or beginning of  the
periods.

<p>Diagram  to  visualize  the  positive  and  negative cash flows (cash flow
diagrams):

<p>Amounts shown above the line are positive, received, and amounts shown below the
line are negative, paid out.

<hr>
<A NAME="Appreciation">
<h2>Appreciation</h2></A>
<br>Appreciation
<br>Depreciation
<br>Compound Growth
<br>Savings Account
<pre>
                                                                A FV*
          1   2   3   4   .   .   .   .   .   .   .   .   .   n |
 Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
        |
        V
        PV
</pre>


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<hr>
<A NAME="Annuity">
<h2>Annuity (series of payments)</h2></A>
<br>Annuity (series of payments)
<br>Pension Fund
<br>Savings Plan
<br>Sinking Fund

<pre>
     PV = 0                                                     A
                                                                |
 Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
        | 1 | 2 | 3 | 4 | . | . | . | . | . | . | . | . | . | n
        V   V   V   V   V   V   V   V   V   V   V   V   V   V
       PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT
</pre>


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<hr>
<A NAME="Amortization">
<h2>Amortization</h2></A>
<br>Direct Reduction Loan
<br>Mortgage (fully amortized)

<pre>
     PV ^
        |                                                      FV=0
 Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
          1 | 2 | 3 | 4 | . | . | . | . | . | . | . | . | . | n |
            V   V   V   V   V   V   V   V   V   V   V   V   V   V
           PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT
</PRE>

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<hr>
<A NAME="Lease">
<H2>Annuity</H2></a>
<br>Annuity
<BR>Lease (with buy back or residual)*
<br>Loan or Mortgage (with balloon)*
<pre>
                                                                A FV*
           PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT PMT  | +
            A   A   A   A   A   A   A   A   A   A   A   A   A   A PMT
          1 | 2 | 3 | 4 | . | . | . | . | . | . | . | . | . | n |
 Period +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
        |
        V
        PV
</pre>

<!--#############################################################################-->

<hr>
<A NAME="Interest">
<h1>Interest</h1></A>
<p>Before discussing the financial equation, we will discuss interest. Most
financial transactions utilize a nominal interest rate, NAR, i.e., the interest
rate per year. The NAR must be converted to the interest rate per payment
period and the compounding accounted for before it can be used in computing
an interest payment. After this conversion process, the interest used is the
effective interest rate, EIR. In converting NAR to EIR, there are two concepts
to discuss first, the Compounding Frequency and the Payment Frequency and
whether the interest is compounded in discrete intervals or continuously.

<A NAME="CompFreq">
<h2>Compounding Frequency</h2></A>
<p>The compounding Frequency, CF, is simply the number of times per year, the
monies in the financial transaction are compounded. In the U.S., monies
are usually compounded daily on bank deposits, and monthly on loans. Somtimes
long term deposits are compounded quarterly or weekly.

<A NAME="PayFreq">
<h2>Payment Frequency</h2></A>
<p>The Payment Frequency, PF, is simply how often during a year payments are
made in the transaction. Payments are usually scheduled on a regular basis
and can be made at the beginning or end of the payment period. If made at
the beginning of the payment period, interest must be applied to the payment
as well as any previous money paid or money still owed.

<A NAME="NorVal">
<h2>Normal CF/PF Values</h2></A>
<p>Normal values for CF and PF are:
<ul>
<ul>
<li>1   == annual
<li>2   == semi-annual
<li>3   == tri-annual
<li>4   == quaterly
<li>6   == bi-monthly
<li>12  == monthly
<li>24  == semi-monthly
<li>26  == bi-weekly
<li>52  == weekly
<li>360 == daily
<li>365 == daily
</ul>
</ul>

<p>The  Compounding  Frequency per year,  CF,  need not be identical  to the
Payment Frequency per year, PF. Also,
Interest may be compounded in either discrete intervals or continuously
compounded and payments may be made at the beginning of the payment period or at the
end of the payment period.

<p>CF and PF are defaulted to  12. The default is for discrete interest intervals
and payments are defaulted to the end of the payment period.

<p>When a solution  for n, PV,  PMT or FV  is required, the  nominal interest
rate, i, must first be converted to the effective interest rate per payment
period. This rate, ieff, is then used to compute the selected variable.  To
convert i to ieff, the following expressions are used:

<A NAME="DiscIntNARtoEIR">
<h2>NAR to EIR for Discrete Interest Periods</h2></A>
<p>To convert NAR to EIR for discrete interest periods:

<p>ieff = (1 + i/CF)^(CF/PF) - 1

<A NAME="ContIntNARtoEIR">
<h2>NAR to EIR for Continuous Compounding</h2></A>
<p>to convert NAR to EIR for Continuous Compounding:

<p>ieff = e^(i/PF) - 1 = exp(i/PF) - 1

<p>When interest is computed, the computation produces the effective interest
rate, ieff. This value must then be converted to the nominal interest rate.
Function  _I  in the <tt>"fin.exp"</tt> utility returns  the  nominal  interest
rate NOT the effective interest rate. ieff is converted to i using the following expressions:

<A NAME="DiscIntEIRtoNAR">
<h2>EIR to NAR for Discrete Interest Periods</h2></A>
<p>To convert EIR to NAR for discrete interest periods:

<p>i = CF*([(1+ieff)^(PF/CF) - 1)

<A NAME="ContIntEIRtoNAR">
<h2>EIR to NAR for Continuous Compounding</h2></A>
<p>To convert EIR to NAR for continuous compounding:

<p>i = ln((1+ieff)^PF)

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<A NAME="FinEquation">
<H1>Financial Equation</h1></A>
<p>NOTE: in the equations below for the financial transaction, all interest rates
are the effective interest rate, <tt>ieff</tt>. The symbol will be shortned to just <tt>i</tt>.

<p>The financial equation used to inter-relate n,i,PV,PMT and FV is:

<p>1) PV*(1 + i)^n + PMT*(1 + iX)*[(1+i)^n - 1]/i + FV = 0

<pre>
   Where: X   == 0 for end of period payments, and
          X   == 1 for beginning of period payments
          n   == number of payment periods
          i   == effective interest rate for payment period
          PV  == Present Value
          PMT == periodic payment
          FV  == Future Value
</pre>

<!--#############################################################################-->
<A NAME="FinDeriv">
<h2>Financial Equation Derivation</h2></A>
<p>The derivation of the financial equation is contained in the
<A HREF="finderv.html#TOP">Financial Equations</A>
section.


<!--#############################################################################-->

<A NAME="AmortSched">
<h1>Amortization Schedules.</h1></A>

<A NAME="ED_IP_Dates">
<h2>Effective and Initial Payment Dates</h2></A>
<p>Financial Transactions have an effective Date, ED, and an Initial Payment
Date, IP. ED may or may not be the same as IP, but IP is always the same
or later than ED. Most financial transaction calculators assume that
IP is equal to ED for beginning of period payments or at the end of the
first payment period for end of period payments.

<p>This is not always true. IP may be delayed for financial reasons such as cash
flow or accounting calendar. The subsequent payments then follow the
agreed upon periodicity.

<A name="Eff_PV">
<h2>Effective Present Value</h2></A>
<p>Since money has a time value, the "delayed" IP
must be accounted for. Computing an "Effective PV", pve, is one means of
handling a delayed IP.

<p>If

<pre>
ED_jdn == the Julian Day Number of ED, and
IP_jdn == the Julian Day Number of IP
</pre>

<p>pve is computed as:

<pre>
   pve = pv*(1 + i)^(s*PF/d*CF)

   Where: d = length of the payment period in days, and
          s = IP_jdn - ED_jdn - d*(1 - X)
</pre>

<A name="iterative_soltn">
<h2>Iterative Amortization Schedule</h2></A>
<p>Computing an amortization Schedule for a given financial transaction is
simply applying the basic equation iteratively for each payment period:

<pre>
   PV[n] = PV[n-1] + (PMT + (PV[n-1] + X * PMT) * i)
         = PV[n-1] * (1 + i) + PMT * (1 + iX)
    for n >= 1
</pre>

<p>At the end of each iteration, PV[n] is rounded to the nearest cent. For
each payment period, the interest due may be computed separately as:

<pre>
   ID[n] = (PV[n-1] + X * PMT) * i
</pre>

<p>and rounded to the nearest cent. PV[n] then becomes:

<pre>
   PV[n] = PV[n-1] + PMT + ID[n]
</pre>

<A name="AnnualSum">
<h2>Annual Summary</h2></A>
<p>For those cases where a yearly summary only is desired, it is not necessary
to compute each transaction for each payment period, rather the PV may be
be computed for the beginning of each year, PV[yr], and the FV computed for
the end of the year, FV[yr]. The interest paid during the year is the computed as:

<pre>
   ID[yr] = (NP * PMT) + PV[yr] + FV[yr]
    where: NP == number of payments during year
              == PF for a full year of payments
</pre>

<A name="FinalPayment">
<h2>Final Payment Calculation</h2></A>
<p>Since the final payment may not be equal to the periodic payment, the final
payment must be computed separately as follows. Two derivations are given below
for the final payment equation. Both derivations are given below since one or
the other may be clearer to some readers. Both derivations are essentially
the same, they just have different starting points. The first is the fastest to derive.

<p>Note, for the purposes of computing an amortization table, the number of periodic
payments is assumed to be an integral value. For most cases this is true, the two
principles in any transaction usually agree upon a certain term or number of periodic
payments. In some calculations, however, this may not hold. In all of the calculations
below, n is assumed integral and in the gnucash implementation, the following calculation
is performed to assure this fact:

<pre>
    n = int(n)
</pre>

<ol>
<li>final_pmt == final payment @ payment n
<p>From the basic financial equation derived above:

<pre>
       PV[n] = PV[n-1]*(1 + i) + final_pmt * (1 + iX), i == effective interest rate
</pre>

<p>solving for final_pmt, we have:
<p>NOTE: FV[n] = -PV[n], for any n

<pre>
       final_pmt * (1 + iX) = PV[n] - PV[n-1]*(1 + i)
                            = FV[n-1]*(1 + i) - FV[n]
       final_pmt = FV[n-1]*(1+i)/(1 + iX) - FV[n]/(1 + iX)

       final_pmt = FV[n-1]*(1 + i) - FV[n], for X == 0, end of period payments
                 = FV[n-1] - FV[n]/(1 + i), for X == 1, beginning of period payments
</pre>

<li>final_pmt == final payment @ payment n

<pre>
       i[n] == interest due @ payment n
       i[n] = (PV[n-1] + X * final_pmt) * i, i == effective interest rate
            = (X * final_pmt - FV[n]) * i
</pre>

<p>Now the final payment is the sum of the interest due, plus the present value
at the next to last payment plus any residual future value after the last payment:

<pre>
       final_pmt = -i[n] - PV[n-1] - FV[n]
                 = FV[n-1] - i[n] - FV[n]
                 = FV[n-1] - (X *final_pmt - FV[n-1])*i - FV[n]
                 = FV[n-1]*(1 + i) - X*final_pmt*i - FV[n]
</pre>

<p>solving for final_pmt:

<pre>
       final_pmt*(1 + iX) = FV[n-1]*(1 + i) - FV[n]
       final_pmt = FV[n-1]*(1 + i)/(1 + iX) - FV[n]/(1 + iX)

       final_pmt = FV[n-1]*(1 + i) - FV[n], for X == 0, end of period payments
                 = FV[n-1] - FV[n]/(1 + i), for X == 1, beginning of period payments
</pre>
</ol>

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<A name="AmortCases">
<h2>Amortization Cases</h2></A>

<p>The amortization schedule is computed for six different situations:

<ol>
<A name="ConstOrigData">
<h3>Constant Repayment to Principal, Original Data</h3>
<li>In a constant repayment to principal loan, each payment varies. A constant amount
is applied to the principal for each payment, usually equal to the originating present value
divided by the number of repayment periods, and the interest for the payment period is
added to the constant principal payment. The derivation of the equation for this type
is contained in the <A HREF="constderv.html#TOP">Constant Repayment Equations</A> section. This
case computes the amortization schedule with the original loan data and a constant repayment
to principal.

<A name="ConstNewData">
<h3>Constant Repayment to Principal, Delayed Repayment</h3>
<li>In a constant repayment to principal loan, each payment varies. A constant amount
is applied to the principal for each payment, usually equal to the originating present value
divided by the number of repayment periods, and the interest for the payment period is
added to the constant principal payment. The derivation of the equation for this type
is contained in the <A HREF="constderv.html#TOP">Constant Repayment Equations</A> section. This
case computes the amortization schedule with the delayed loan data and a constant repayment
to principal.

<A name="OrigData">
<h3>Original Data Schedule</h3></A>
<li>The original financial data is used. This ignores any possible agjustment to
the Present value due to any delay in the initial payment. This is quite
common in mortgages where end of period payments are used and the first
payment is scheduled for the end of the first whole period, i.e., any
partial payment period from ED to the beginning of the next payment period
is ignored.

<A name="NewFinalPayment">
<h3>Recomputed Final Payment</h3></A>
<li>The original periodic payment is used, the Present Value is adjusted for the
delayed Initial Payment. The total number of payments remains the same. The
final payment is adjusted to bring the balance into agreement with the
agreed upon final Future Value.

<A name="NewPayment">
<h3>Recomputed Periodic Payment</h3></A>
<li>A new periodic payment is computed based upon the adjusted Present Value, the
originally agreed upon number of total payments and the agreed upon Future Value.
The new periodic payments are computed to minimize the final payment in accordance
with the Future Value after the last payment.

<A name="NewTerm">
<h3>Recomputed Term</h3></A>
<li>The original periodic payment is retained and a new number of total payments is computed
based upon the adjusted Present Value and the agreed upon Future Value.
</ol>

<a name="DisplaySched">
<h2>Amortization Schedule Display</h2></A>
<p>The amortization schedule may be computed and displayed in three manners:

<ol>
<li>The payment#, interest paid, principal paid and remaining PV for each payment period
are computed and displayed.
<p>At the end of each year a summary is computed and displayed
and the total interest paid is diplayed at the end.

<li>A summary is computed and displayed for each year. The interest paid during the
year is computed and displayed as well as the remaining balance at years end.
<p>The total interest paid is diplayed at the end.

<li>An amortization schedule is computed and displayed for a common method of
advanced payment of principal.
<p>In this amortization schedule, the principal for the
next payment is computed and added into the current payment. This method will
cut the number of total payments in half and will cut the interest paid almost
in half.
<p>For mortgages, this method of prepayment has the advantage of keeping
the total payments small during the initial payment periods
The payments grow until the last payment period when presumably the borrower
can afford larger payments.
</ol>

<!--================================================================================-->
<p>NOTE: For Payment Frequencies, PF, semi-monthly or less, i.e., PF == 12 or PF == 24,
a 360 day calendar year and 30 day month are used. For Payment Frequencies, PF,
greater than semi-monthly, PF > 24, the actual number of days per year and per payment
period are used. The actual values are computed using the built-in 'jdn' function

<!--#############################################################################-->
<a name="Usage">
<h1>Financial Calculator Usage</h1></a>
<p>the Financial Calculator is run as a QTAwk utility. If input is to be interactive and
from the keyboard, do not specify any input files on the command line. The financial
calcutlator reads all input from the standard input file. The calculator is started
as:

<pre>
QTAwk -f fin.exp
</pre>

<p>The calculator will clear the display screen and display a two screen help:

<pre>
Financial Calculator
Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
To compute Loan Quantities:
N ==> to compute # payment periods from i, pv, pmt, fv
_N(i,pv,pmt,fv,CF,PF,disc,bep) ==> to compute # payment periods
I ==> to compute nominal interest rate from n, pv, pmt, fv, CF, PF, disc, bep
_I(n,pv,pmt,fv,CF,PF,disc,bep) ==> to compute interest
PV ==> to compute Present Value from n, i, pmt, fv, CF, PF, disc, bep
_PV(n,i,pmt,fv) ==> to compute Present Value
PMT ==> to compute Payment from n, i, pv, fv, CF, PF, disc, bep
_PMT(n,i,pv,fv,CF,PF,disc,bep) ==> to compute Payment
FV ==> to compute Future Value from n, i, pv, pmt, CF, PF, disc, bep
_FV(n,i,pv,pmt,CF,PF,disc,bep) ==> to compute Future Value
Press Any Key to Continue
</pre>

<p>The first screen displays the calculator commands which are available. Press any key
and a second screen displays the variables defined by the calculator and which must be
set by the user to use the financial calculator functions.

<pre>
[Aa](mort)? to Compute Amortization Schedule
[Cc](ls)? to Clear Screen
[Dd](efault)? to Re-Initialize
[Hh](elp) to Display This Help
[Qq](uit)? to Quit
[Ss](tatus)? to Display Status of Computations
[Uu](ser) Display User Defined Variables

Variables to set:
n    == number of periodic payments
i    == interest per compouding interval
pv   == present value
pmt  == periodic payment
fv   == future value
disc == TRUE/FALSE == discrete/continuous compounding
bep  == TRUE/FALSE == beginning of period/end of period payments
CF   == compounding frequency per year
PF   == payment frequency per year

ED   == effective date of transaction, mm/dd/yyyy
IP   == initial payment date of transaction, mm/dd/yyyy
</pre>

<a name="FinCommands">
<h2>Calculator Commands</h2></a>
<p>The financial calculator commands available are listed above and below.

<p>Note that the first letter of the command is all that is necessary to activate the
desired function.

<ol>
<li>[Aa](mort)? to Compute Amortization Schedule
<br>After all financial variables have been defined as well as the transaction dates,
the amortization schedule can be computed for all financial transactions in which
one would make sense.
<li>[Cc](ls)? to Clear Screen
<br>This command clears the screen and displays the copyright.
<li>[Dd](efault)? to Re-Initialize
<br>This command re-initializes all calculator variables to their start-up values.
<li>[Hh](elp) to Display This Help
<br>This command is used to display the start-up help screens at any time.
<li>[Qq](uit)? to Quit
<br>When the calculator is used interactively from the keyboard, this command allows
the user to terminate the calculator session.
<li>[Ss](tatus)? to Display Status of Computations
<br>This command displays the status of the calculator variables. A typical status display
would be:

<pre>
Financial Calculator
Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved
Current Financial Calculator Status:
Compounding Frequency: (CF) 12
Payment     Frequency: (PF) 12
Compounding: Discrete (disc = TRUE)
Payments: End of Period (bep = FALSE)
Number of Payment Periods (n): 360              (Years: 30)
Nominal Annual Interest Rate (i): 7.25
  Effective Interest Rate Per Payment Period: 0.00604167
Present Value (pv): 233,350.00
Periodic Payment (pmt): -1,591.86
Future Value (fv): 0.00
Effective       Date: Tue Jun 04 00:00:00 1996(2450239)
Initial Payment Date: Thu Aug 01 00:00:00 1996(2450297)
<>
</pre>
<li>[Uu](ser) Display User Defined Variables
<br>If any variables have been defined by the user, this command displays their names and
values.
</ol>

<a name="CalcInput">
<h2>Calculator Input</h2></a>
<p>The calculator displays an <tt>input prompt</tt> whenever it is waiting for input
from the keyboard. The <tt>input prompt</tt> is simply <tt>&#60;&#62;</tt>. The desired
input is typed at the keyboard and the enter key pressed. The result of calculating the
value of the input line is then displayed by the calculator. For example, if the user wanted
to set the value of the nominal interest in the calculator to 6.25, the following line would be
input to the calculator:

<p><tt>i=6.25</tt>.

<p>A semi-colon at the end of the input is optional.
The line as seen on the display with the calculator input prompt would be:

<pre>
<>i = 6.25
    6.25
</pre>

<p>Note that the calculator displays the value of the result, 6.25 in this case.

<p>The calculator is controlled by setting the calculator variables to the desired values
and <tt>"executing"</tt> the calculator functions to derive the values for the unknown
variables. For example, for a conventional home mortgage for $233,350.00 with a thirty year
term, nominal annual rate of 7.25%, n, i, pv and fv are known:

<pre>
n == 360 == 12 * 30
i == 7.25
pv= 233350
fv = 0
</pre>

<p>The payments to completely pay off the mortgage with the 360 periodic payments is desired.
To compute the desired periodic payment value, the <tt>PMT</tt> function is used. Since the
function has no defined arguments, in invoking the function no arguments are specified. The
complete session to input the desired values and calculate the periodic payment value would
appear as:

<pre>
<>n=30*12
        360
<>i=7.25
        7.25
<>pv=233350
        233,350
<>PMT
        -1,591.86
</pre>

<p>Note that the input may contain computations, <tt>n=30*12</tt>. In addition, any QTAwk
built-in function may be specified and any functions defined in the financial calculator.
This can be handy for computing intermediate values or other results from the results of
the calculator.

<p>Note that the output of the <tt>PMT</tt> function is rounded to the nearest cent. Over the
thirty year term of the payback, the rounding will affect the last payment. To determine
the balance due, fv, after 359 payment have been made, decrement n by 1 and compute the
future value:

<pre>
<>n-=1
        359
<>FV
        -1,580.20
<>n+=1
        360
<>FV
        2.12
<>
</pre>

<p>The future value after 359 payments is less than the periodic payment and a full final payment
will overpay the loan. The final FV computation with n restored to 360 shows an overpayment
of 2.12.

<a name="CalcFun">
<h2>Calculator Functions</h2></a>
<p>The calculator functions:

<pre>
N
I
PV
PMT
FV
</pre>

<p>can be used to calculate the variable with the corresponding lower case name, using the
values of the other four calculator variables which have already been set. In addition, the
calculator functions:

<pre>
_N(i,pv,pmt,fv,CF,PF,disc,bep)
_I(n,pv,pmt,fv,CF,PF,disc,bep)
_PV(n,i,pmt,fv,CF,PF,disc,bep)
_PMT(n,i,pv,fv,CF,PF,disc,bep)
_FV(n,i,pv,pmt,CF,PF,disc,bep)
</pre>

<p>can be used to compute the value of the corresponding quantity for any specified value
of the input arguments.

<p>There are three differences between the functions <tt>N, I, PV, PMT, FV</tt> and the
functions
<tt>_N(i,pv,pmt,fv,CF,PF,disc,bep), _I(n,pv,pmt,fv,CF,PF,disc,bep), _PV(n,i,pmt,fv,CF,PF,disc,bep),
_PMT(n,i,pv,fv,CF,PF,disc,bep), _FV(n,i,pv,pmt,CF,PF,disc,bep)</tt>.
<ol>
<li>The first set of functions take no arguments and
use the calculator variables, n, i, pv, pmt, fv, CF, PF, disc
and bep to compute the desired value. The second set of functions use the values passed in
the function arguments. The first set of functions call the second set with the necssary
arguments.
<li>The first set of functions round the computed value returned by the call to the second set
of functions to the nearest cent. The second set of functions perform no rounding.
<li>The first set of functions set the calculator variables with the corresponding lower case name
to the value computed. The second set of functions set no global variable values.
</ol>

<a name="UserVar">
<h2>User Defined Variables</h2></a>
<p>User defined variables may be defined and their values set to a desired qunatity. For example,
to save computation results before re-initializing the calculator to obtain other results. If
the user desired to compare the periodic payments necessary to fully pay the conventional
mortgage cited above, the payment computed above could be saved in the variable
<tt>end_pmt</tt>, the payments set to beginning of period payments and the new payment
computed. The new value could be set into the variable <tt>beg_pmt</tt>. The two payments
could then be viewed with the <tt>u</tt> command. The difference could then be computed
between the two payment methods:

<pre>
<>n=30*12
        360
<>i=7.25
        7.25
<>pv=233350
        233,350
<>PMT
        -1,591.86
<>end_pmt=pmt
        -1,591.86
<>bep=1
        1
<>PMT
        -1,582.30
<>beg_pmt=pmt
        -1,582.30
<>u

Financial Calculator
Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
Current Financial Calculator Status:
User Defined Variables:
end_pmt == -1,591.86
beg_pmt == -1,582.30
<>beg_pmt-end_pmt
        9.56
<>
</pre>

<p>The financial calculator is thus a true calculator and can be used for computations
desired by the user beyond those performed by the functions of the utility.

<a name="Rounding">
<h2>Rounding</h2></a>
<p>Note that the output of the calculator is rounded to the nearest cent for floating
point values. Sometimes the full accuracy of the value is desired. This can be obtained
by redefing the calculator variable <tt>ofmt</tt> to the string "%.15g". You might want to
save the current value in a user variable for resetting. For example in the above
conventional mortgage, the exact value of the periodic payment can be displayed as:

<pre>
<>sofmt=ofmt
        "%.2f"
<>ofmt="%.15g"
        "%.15g"
<>pmt=_PMT(n,i,pv,fv,CF,PF,disc,bep)
        -1,591.85834951112
<>ofmt=sofmt
        "%.2f"
<>
</pre>

<p>Note that the current value of the output format string, <tt>ofmt</tt>, has been
saved in the variable, <tt>sofmt</tt>, and later restored.

<a name="Examples">
<h1>Examples</h1></a>
<!-- Note: in the following examples, the user input is preceded by the prompt     -->
<!-- "<>". The result of evaluating the input expression is then displayed.         -->
<!-- I have taken the liberty of including comments in the example                  -->
<!-- input/output sessions by preceding with '#'. Thus, for the line:              -->
<!-- <>n=5         #set number of periods                                           -->
<!-- the comment that setting the number of periods is not really input and the true-->
<!-- input is only:                                                                 -->
<!-- <>n=5                                                                          -->

<a name="SimpleInt">
<h2>Simple Interest </h2></a>
<p>Simple Interest
<p> Find the annual simple interest rate (%) for an $800 loan to be repayed at the
 end of one year with a single payment of $896.
<pre>
 <>d
 <>CF=PF=1
         1
 <>n=1
         1
 <>pv=-800
         -800
 <>fv=896
         896
 <>I
         12.00
</pre>

<a name="CompundInt">
<h2>Compound Interest</h2></a>
<p>Compound Interest
<p>Find the future value of $800 after one year at a nominal rate of 12%
 compounded monthly. No payments are specified, so the payment frequency is
 set equal to the compounding frequency at the default values.
<pre>
 <>d
 <>n=12
         12
 <>i=12
         12
 <>pv=-800
         -800
 <>FV
          901.46
</pre>

<a name="PeriodicPmt">
<h2>Periodic Payment</h2></a>
<p>Periodic Payment
<p>Find the monthly end-of-period payment required to fully amortize the loan
 in Example 2. A fully amortized loan has a future value of zero.
<pre>
 <>fv=0
        0
 <>PMT
        71.08
</pre>

<a name="ConvMortg">
<h2>Conventional Mortgage</h2></a>
<p>Conventional Mortgage
<p>Find the number of monthly payments necessary to fully amortize a loan of
 $100,000 at a nominal rate of 13.25% compounded monthly, if monthly end-of-period
 payments of $1125.75 are made.
<pre>
 <>d
 <>i=13.25
         13.25
 <>pv=100000
         100,000
 <>pmt=-1125.75
         -1,125.75
 <>N
         360.10
</pre>

<a name="FinalPmt">
<h2>Final Payment</h2></a>
<p>Final Payment
<p>Using the data in the above example, find the amount of the final payment if n is
changed to 360. The final payment will be equal to the regular payment plus
any balance, future value, remaining at the end of period number 360.
<pre>
 <>n=int(n)
        360
 <>FV
        -108.87
 <>pmt+fv
        -1,234.62
</pre>

<a name="AS_AnnualSum">
<h2>Conventional Mortgage Amortization Schedule - Annual Summary</h2></a>
<p>Conventional Mortgage Amortization Schedule - Annual Summary
<p>Using the data from the loan in the previous example, compute the amortization
schedule when the
Effective date of the loan is June 6, 1996 and the initial payment is
made on August 1, 1996. Ignore any change in the PV due to the delayed
initial payment caused by the partial payment period from June 6 to July 1.

<pre>
 <>ED=6/6/1996
 Effective Date set: (2450241) Thu Jun 06 00:00:00 1996
 <>IP=8/1/96
 Initial Payment Date set: (2450297) Thu Aug 01 00:00:00 1996
 <>a
   Effective       Date: Thu Jun 06 00:00:00 1996
   Initial Payment Date: Thu Aug 01 00:00:00 1996
   The amortization options are:
   The Old Present Value (pv)     was: 100,000.00
   The Old Periodic Payment (pmt) was: -1,125.75
   The Old Future  Value (fv)     was: -108.87
   1: Amortize with Original Transaction Values
       and final payment: -1,125.75

   The New Present Value (pve)  is:  100,919.30
   The New Periodic Payment (pmt) is:  -1,136.10
   2: Amortize with Original Periodic Payment
       and final payment: -49,023.68
   3: Amortize with New Periodic Payment
       and final payment: -1,132.57
   4: Amortize with Original Periodic Payment,
       new number of total payments (n): 417
       and final payment: -2,090.27

   Enter choice 1, 2, 3 or 4: <>
</pre>

<p>Press '1' to choose option 1:

<pre>
    Amortization Schedule:
   Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
   Enter choice y, p or a:
   <>
</pre>

<p>Press 'y' for an annual summary:

<pre>
   Enter Filename for Amortization Schedule.
     (null string uses Standard Output):
</pre>

<p>Press enter to display output on screen:

<pre>
  Amortization Table
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  Compounding Frequency per year: 12
  Payment     Frequency per year: 12
  Compounding: Discrete
  Payments: End of Period
  Payments (359): -1,125.75
  Final payment (# 360): -1,125.75
  Nominal Annual Interest Rate: 13.25
    Effective Interest Rate Per Payment Period: 0.0110417
  Present Value: 100,000.00
  Year      Interest   Ending Balance
  1996     -5,518.42       -99,889.67
  1997    -13,218.14       -99,598.81
  1998    -13,177.17       -99,266.98
  1999    -13,130.43       -98,888.41
  2000    -13,077.11       -98,456.52
  2001    -13,016.28       -97,963.80
  2002    -12,946.88       -97,401.68
  2003    -12,867.70       -96,760.38
  2004    -12,777.38       -96,028.76
  2005    -12,674.33       -95,194.09
  2006    -12,556.76       -94,241.85
  2007    -12,422.64       -93,155.49
  2008    -12,269.63       -91,916.12
  2009    -12,095.06       -90,502.18
  2010    -11,895.91       -88,889.09
  2011    -11,668.70       -87,048.79
  2012    -11,409.50       -84,949.29
  2013    -11,113.78       -82,554.07
  2014    -10,776.41       -79,821.48
  2015    -10,391.53       -76,704.01
  2016     -9,952.43       -73,147.44
  2017     -9,451.49       -69,089.93
  2018     -8,879.99       -64,460.92
  2019     -8,227.99       -59,179.91
  2020     -7,484.16       -53,155.07
  2021     -6,635.56       -46,281.63
  2022     -5,667.43       -38,440.06
  2023     -4,562.94       -29,494.00
  2024     -3,302.89       -19,287.89
  2025     -1,865.36        -7,644.25
  2026       -236.00          -108.87

  Total Interest: -305,270.00
</pre>

<p> NOTE: The amortization table leaves the FV as it was when the amortization
function was entered. Thus, a balance of 108.87 is due at the end of the
table. To completely pay the loan, set fv to 0.0:
<pre>
<>fv=0
    0
<>a
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  The amortization options are:
  The Old Present Value (pv)     was: 100,000.00
  The Old Periodic Payment (pmt) was: -1,125.75
  The Old Future  Value (fv)     was: 0.00
  1: Amortize with Original Transaction Values
      and final payment: -1,234.62

  The New Present Value (pve)  is:  100,919.30
  The New Periodic Payment (pmt) is:  -1,136.12
  2: Amortize with Original Periodic Payment
      and final payment: -49,132.55
  3: Amortize with New Periodic Payment
      and final payment: -1,148.90
  4: Amortize with Original Periodic Payment,
      new number of total payments (n): 417
      and final payment: -2,199.14

  Enter choice 1, 2, 3 or 4: <>
</pre>

<p>Press '1' for option 1:

<pre>
    Amortization Schedule:
   Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
   Enter choice y, p or a:
   <>
</pre>

<p>Press 'y' for annual summary:

<pre>
   Enter Filename for Amortization Schedule.
     (null string uses Standard Output):
</pre>

<p>Press enter to display output on screen:

<pre>
  Amortization Table
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  Compounding Frequency per year: 12
  Payment     Frequency per year: 12
  Compounding: Discrete
  Payments: End of Period
  Payments (359): -1,125.75
  Final payment (# 360): -1,234.62
  Nominal Annual Interest Rate: 13.25
    Effective Interest Rate Per Payment Period: 0.0110417
  Present Value: 100,000.00
  Year      Interest   Ending Balance
  1996     -5,518.42       -99,889.67
  1997    -13,218.14       -99,598.81
  1998    -13,177.17       -99,266.98
  1999    -13,130.43       -98,888.41
  2000    -13,077.11       -98,456.52
  2001    -13,016.28       -97,963.80
  2002    -12,946.88       -97,401.68
  2003    -12,867.70       -96,760.38
  2004    -12,777.38       -96,028.76
  2005    -12,674.33       -95,194.09
  2006    -12,556.76       -94,241.85
  2007    -12,422.64       -93,155.49
  2008    -12,269.63       -91,916.12
  2009    -12,095.06       -90,502.18
  2010    -11,895.91       -88,889.09
  2011    -11,668.70       -87,048.79
  2012    -11,409.50       -84,949.29
  2013    -11,113.78       -82,554.07
  2014    -10,776.41       -79,821.48
  2015    -10,391.53       -76,704.01
  2016     -9,952.43       -73,147.44
  2017     -9,451.49       -69,089.93
  2018     -8,879.99       -64,460.92
  2019     -8,227.99       -59,179.91
  2020     -7,484.16       -53,155.07
  2021     -6,635.56       -46,281.63
  2022     -5,667.43       -38,440.06
  2023     -4,562.94       -29,494.00
  2024     -3,302.89       -19,287.89
  2025     -1,865.36        -7,644.25
  2026       -344.87             0.00

  Total Interest: -305,378.87
</pre>

<p>Note that now the final payment differs from the periodic payment and
the loan has been fully paid off.

<a name="AS_PeriodicPmt">
<h2>Conventional Mortgage Amortization Schedule - Periodic Payment Schedule</h2></a>
<p>Conventional Mortgage Amortization Schedule  - Periodic Payment Schedule
<p>Using the loan in the previous example, compute the amortization table and display the
results for each payment period.
As in example 6, ignore any increase in the PV due to the
delayed IP.

<pre>
<>
  Amortization Table
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  Compounding Frequency per year: 12
  Payment     Frequency per year: 12
  Compounding: Discrete
  Payments: End of Period
  Payments (359): -1,125.75
  Final payment (# 360): -1,234.62
  Nominal Annual Interest Rate: 13.25
    Effective Interest Rate Per Payment Period: 0.0110417
  Present Value: 100,000.00
  Pmt#       Interest      Principal        Balance
     1      -1,104.17         -21.58     -99,978.42
     2      -1,103.93         -21.82     -99,956.60
     3      -1,103.69         -22.06     -99,934.54
     4      -1,103.44         -22.31     -99,912.23
     5      -1,103.20         -22.55     -99,889.68
  Summary for 1996:
    Interest  Paid: -5,518.43
    Principal Paid: -110.32
    Year Ending Balance: -99,889.68
    Sum of Interest Paid: -5,518.43
  Pmt#       Interest      Principal        Balance
     6      -1,102.95         -22.80     -99,866.88
     7      -1,102.70         -23.05     -99,843.83
     8      -1,102.44         -23.31     -99,820.52
     9      -1,102.18         -23.57     -99,796.95
    10      -1,101.92         -23.83     -99,773.12
    11      -1,101.66         -24.09     -99,749.03
    12      -1,101.40         -24.35     -99,724.68
    13      -1,101.13         -24.62     -99,700.06
    14      -1,100.85         -24.90     -99,675.16
    15      -1,100.58         -25.17     -99,649.99
    16      -1,100.30         -25.45     -99,624.54
    17      -1,100.02         -25.73     -99,598.81
  Summary for 1997:
    Interest  Paid: -13,218.13
    Principal Paid: -290.87
    Year Ending Balance: -99,598.81
    Sum of Interest Paid: -18,736.56
  Pmt#       Interest      Principal        Balance
    18      -1,099.74         -26.01     -99,572.80
    19      -1,099.45         -26.30     -99,546.50
    .
    .
    .
   346        -171.99        -953.76     -14,622.84
   347        -161.46        -964.29     -13,658.55
   348        -150.81        -974.94     -12,683.61
   349        -140.05        -985.70     -11,697.91
   350        -129.16        -996.59     -10,701.32
   351        -118.16      -1,007.59      -9,693.73
   352        -107.03      -1,018.72      -8,675.01
   353         -95.79      -1,029.96      -7,645.05
  Summary for 2025:
    Interest  Paid: -1,865.45
    Principal Paid: -11,643.55
    Year Ending Balance: -7,645.05
    Sum of Interest Paid: -305,034.80
  Pmt#       Interest      Principal        Balance
   354         -84.41      -1,041.34      -6,603.71
   355         -72.92      -1,052.83      -5,550.88
   356         -61.29      -1,064.46      -4,486.42
   357         -49.54      -1,076.21      -3,410.21
   358         -37.65      -1,088.10      -2,322.11
   359         -25.64      -1,100.11      -1,222.00
  Final Payment (360): -1,235.49
   360         -13.49      -1,222.00           0.00
  Summary for 2026:
    Interest  Paid: -344.94
    Principal Paid: -7,645.05

  Total Interest: -305,379.74
</pre>

<p>The complete amortization table can be viewed in the
<A HREF="./amortp.html#AmortPer">Periodic Amortization Schedule</A> for this loan.

<p>You will notice several differences between this amortization schedule and the
Annual Summary Schedule. The Periodic Payment Schedule lists the interest paid for
each payment as well as the principal paid and the remaining balance to be repaid.
At the end of each year an annual summary is printed. At the end of the table the
total interest is printed as in the Annual Summary Schedule.

<p>You will notice that the total interest output at the end of the Periodic Payment
Schedule differs slightly from the total interest output at the end of the Annual Summary
Schedule:

<p>Total Interest for Periodic Payment Schedule:
<pre>
  Total Interest: -305,379.74
</pre>

<p>Total Interest for Annual Summary Schedule:

<pre>
  Total Interest: -305,378.87
</pre>

<p>The difference in total interest is due to the rounding of all quantities at
each periodic payment. The Total Interest paid shown in the Periodic Payment
Schedule will be the more accurate since all quantities exchanged in a financial
transaction will be done to the nearest cent.

<a name="AS_VarAdvPmt">
<h2>Conventional Mortgage Schedule - Variable Advanced Payments</h2></a>
<p>Conventional Mortgage Schedule - Variable Advanced Payments
<p>Again using the loan in the previous examples, compute the amortization table using
the advanced payment
option to prepay the loan. As in the previous example, ignore any increase in the PV due to the
delayed IP.

<pre>
<a>
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  The amortization options are:
  The Old Present Value (pv)     was: 100,000.00
  The Old Periodic Payment (pmt) was: -1,125.75
  The Old Future  Value (fv)     was: 0.00
  1: Amortize with Original Transaction Values
      and final payment: -1,234.62

  The New Present Value (pve)  is:  100,919.30
  The New Periodic Payment (pmt) is:  -1,136.12
  2: Amortize with Original Periodic Payment
      and final payment: -49,132.55
  3: Amortize with New Periodic Payment
      and final payment: -1,148.90
  4: Amortize with Original Periodic Payment,
      new number of total payments (n): 417
      and final payment: -2,199.14

  Enter choice 1, 2, 3 or 4: <>
</pre>

<p>Press 1 for option 1:

<pre>
   Amortization Schedule:
  Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
  Enter choice y, p or a:
  <>
</pre>

<p>Press a for the Advanced Payment Option:

<pre>
  Enter Filename for Amortization Schedule.
    (null string uses Standard Output):
</pre>

<p>Press enter to display output on screen:

<pre>
  Amortization Table
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  Compounding Frequency per year: 12
  Payment     Frequency per year: 12
  Compounding: Discrete
  Payments: End of Period
  Payments (359): -1,125.75
  Final payment (# 360): -1,234.62
  Nominal Annual Interest Rate: 13.25
    Effective Interest Rate Per Payment Period: 0.0110417
  Present Value: 100,000.00
  Advanced Prepayment Amortization
  Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
     1    -1,104.17       -21.58       -21.82    -1,147.57   -99,956.60
     2    -1,103.69       -22.06       -22.31    -1,148.06   -99,912.23
     3    -1,103.20       -22.55       -22.80    -1,148.55   -99,866.88
     4    -1,102.70       -23.05       -23.31    -1,149.06   -99,820.52
     5    -1,102.18       -23.57       -23.83    -1,149.58   -99,773.12
  Summary for 1996:
    Interest  Paid: -5,515.94
    Principal Paid: -226.88
    Year Ending Balance: -99,773.12
    Sum of Interest Paid: -5,515.94
  Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
     6    -1,101.66       -24.09       -24.35    -1,150.10   -99,724.68
     7    -1,101.13       -24.62       -24.90    -1,150.65   -99,675.16
     8    -1,100.58       -25.17       -25.45    -1,151.20   -99,624.54
     9    -1,100.02       -25.73       -26.01    -1,151.76   -99,572.80
    10    -1,099.45       -26.30       -26.59    -1,152.34   -99,519.91
    11    -1,098.87       -26.88       -27.18    -1,152.93   -99,465.85
    12    -1,098.27       -27.48       -27.78    -1,153.53   -99,410.59
    13    -1,097.66       -28.09       -28.40    -1,154.15   -99,354.10
    14    -1,097.03       -28.72       -29.03    -1,154.78   -99,296.35
    15    -1,096.40       -29.35       -29.68    -1,155.43   -99,237.32
    16    -1,095.75       -30.00       -30.34    -1,156.09   -99,176.98
    17    -1,095.08       -30.67       -31.01    -1,156.76   -99,115.30
  Summary for 1997:
    Interest  Paid: -13,181.90
    Principal Paid: -657.82
    Year Ending Balance: -99,115.30
    Sum of Interest Paid: -18,697.84
  Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
    18    -1,094.40       -31.35       -31.70    -1,157.45   -99,052.25
    19    -1,093.70       -32.05       -32.40    -1,158.15   -98,987.80
    20    -1,092.99       -32.76       -33.12    -1,158.87   -98,921.92
    .
    .
    .
   167      -298.87      -826.88      -836.01    -1,961.76   -25,404.90
   168      -280.51      -845.24      -854.57    -1,980.32   -23,705.09
   169      -261.74      -864.01      -873.55    -1,999.30   -21,967.53
   170      -242.56      -883.19      -892.94    -2,018.69   -20,191.40
   171      -222.95      -902.80      -912.77    -2,038.52   -18,375.83
   172      -202.90      -922.85      -933.04    -2,058.79   -16,519.94
   173      -182.41      -943.34      -953.76    -2,079.51   -14,622.84
  Summary for 2010:
    Interest  Paid: -3,448.07
    Principal Paid: -20,232.96
    Year Ending Balance: -14,622.84
    Sum of Interest Paid: -152,300.57
  Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
   174      -161.46      -964.29      -974.94    -2,100.69   -12,683.61
   175      -140.05      -985.70      -996.59    -2,122.34   -10,701.32
   176      -118.16    -1,007.59    -1,018.72    -2,144.47    -8,675.01
   177       -95.79    -1,029.96    -1,041.34    -2,167.09    -6,603.71
   178       -72.92    -1,052.83    -1,064.46    -2,190.21    -4,486.42
   179       -49.54    -1,076.21    -1,088.10    -2,213.85    -2,322.11
   180       -25.64    -1,100.11    -1,222.00    -2,347.75         0.00
  Summary for 2011:
    Interest  Paid: -663.56
    Principal Paid: -14,622.84

  Total Interest: -152,964.13
</pre>

<p>The complete amortization table can be viewed in the
<A HREF="./amorta.html#AmortAdv">Advanced Payment Amortization Schedule</A> for this loan.

<p>This schedule has added two columns over the Periodic Payment Schedule in Example 7. Namely,
<tt>Prepay</tt> and the <tt>Total Pmt</tt> columns. The <tt>Prepay</tt> column is the
amount of the loan prepayment for the period. The <tt>Total Pmt</tt> column is the sum
of the periodic payment and the Prepayment. Note that both the <tt>Prepay</tt> and the
<tt>Total Pmt</tt> quantities increase with each period.

<a name="AS_ConstAdvPmt">
<h2>Conventional Mortgage Schedule - Constant Advanced Payments</h2></a>
<p>Conventional Mortgage Schedule - Constant Advanced Payments
<p>Using the loan in the previous examples, compute the amortization table using
another payment option for repaying a loan ahead of schedule and reducing the interest
paid, constant repayments at each periodic payment. Suppose a constant $100.00 is paid
towards the principal with each periodic payment. How many payments are needed to fully payoff
the loan and what is the total interest paid?

<p>As in the previous example, ignore any increase in the PV due to the
delayed IP.

<p>There are two ways to compute the amortization table for this type of prepayment option.
In the first method, set the variable 'FP' to the amount of the monthly prepayment.

<pre>
<>FP=-100
  -100
<>a
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  The amortization options are:
  The Old Present Value (pv)     was: 100,000.00
  The Old Periodic Payment (pmt) was: -1,125.75
  The Old Future  Value (fv)     was: 0.00
  1: Amortize with Original Transaction Values
      and final payment: -1,234.62

  The New Present Value (pve)  is:  100,919.30
  The New Periodic Payment (pmt) is:  -1,136.12
  2: Amortize with Original Periodic Payment
      and final payment: -49,132.55
  3: Amortize with New Periodic Payment
      and final payment: -1,148.90
  4: Amortize with Original Periodic Payment,
      new number of total payments (n): 417
      and final payment: -2,199.14

  Enter choice 1, 2, 3 or 4: <>
</pre>

<p>Press 1 for option 1:

<pre>
   Amortization Schedule:
  Yearly, y, per Payment, p, Advanced Payment, a, or Fixed Prepayment, f, Amortization
  Enter choice y, p, a or f:
  <>
</pre>

<p>Press f for the Fixed Prepayment schedule.

<pre>
  Enter Filename for Amortization Schedule.
    (null string uses Standard Output):
</pre>

<p>Press enter to display output on screen:

<pre>
Amortization Table
Effective       Date: Thu Jun  6 00:00:00 1996
Initial Payment Date: Thu Aug  1 00:00:00 1996
Compounding Frequency per year: 12
Payment     Frequency per year: 12
Compounding: Discrete
Payments: End of Period
Payments (359): -1,125.75
Final payment (# 360): -1,234.62
Nominal Annual Interest Rate: 13.25
  Effective Interest Rate Per Payment Period: 0.0110417
Present Value: 100,000.00
Advanced Prepayment Amortization - fixed prepayment: -100.00
Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
   1    -1,104.17       -21.58      -100.00    -1,225.75   -99,878.42
   2    -1,102.82       -22.93      -100.00    -1,225.75   -99,755.49
   3    -1,101.47       -24.28      -100.00    -1,225.75   -99,631.21
   4    -1,100.09       -25.66      -100.00    -1,225.75   -99,505.55
   5    -1,098.71       -27.04      -100.00    -1,225.75   -99,378.51
Summary for 1996:
  Interest  Paid: -5,507.26
  Principal Paid: -621.49
  Year Ending Balance: -99,378.51
  Sum of Interest Paid: -5,507.26
Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
   6    -1,097.30       -28.45      -100.00    -1,225.75   -99,250.06
   7    -1,095.89       -29.86      -100.00    -1,225.75   -99,120.20
   8    -1,094.45       -31.30      -100.00    -1,225.75   -98,988.90
   9    -1,093.00       -32.75      -100.00    -1,225.75   -98,856.15
  10    -1,091.54       -34.21      -100.00    -1,225.75   -98,721.94
  11    -1,090.05       -35.70      -100.00    -1,225.75   -98,586.24
  12    -1,088.56       -37.19      -100.00    -1,225.75   -98,449.05
  13    -1,087.04       -38.71      -100.00    -1,225.75   -98,310.34
  14    -1,085.51       -40.24      -100.00    -1,225.75   -98,170.10
  15    -1,083.96       -41.79      -100.00    -1,225.75   -98,028.31
  16    -1,082.40       -43.35      -100.00    -1,225.75   -97,884.96
  17    -1,080.81       -44.94      -100.00    -1,225.75   -97,740.02
Summary for 1997:
  Interest  Paid: -13,070.51
  Principal Paid: -1,638.49
  Year Ending Balance: -97,740.02
  Sum of Interest Paid: -18,577.77
.
.
.

Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
 186      -298.60      -827.15      -100.00    -1,225.75   -26,115.84
 187      -288.36      -837.39      -100.00    -1,225.75   -25,178.45
 188      -278.01      -847.74      -100.00    -1,225.75   -24,230.71
 189      -267.55      -858.20      -100.00    -1,225.75   -23,272.51
 190      -256.97      -868.78      -100.00    -1,225.75   -22,303.73
 191      -246.27      -879.48      -100.00    -1,225.75   -21,324.25
 192      -235.46      -890.29      -100.00    -1,225.75   -20,333.96
 193      -224.52      -901.23      -100.00    -1,225.75   -19,332.73
 194      -213.47      -912.28      -100.00    -1,225.75   -18,320.45
 195      -202.29      -923.46      -100.00    -1,225.75   -17,296.99
 196      -190.99      -934.76      -100.00    -1,225.75   -16,262.23
 197      -179.56      -946.19      -100.00    -1,225.75   -15,216.04
Summary for 2012:
  Interest  Paid: -2,882.05
  Principal Paid: -11,826.95
  Year Ending Balance: -15,216.04
  Sum of Interest Paid: -156,688.79
Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
 198      -168.01      -957.74      -100.00    -1,225.75   -14,158.30
 199      -156.33      -969.42      -100.00    -1,225.75   -13,088.88
 200      -144.52      -981.23      -100.00    -1,225.75   -12,007.65
 201      -132.58      -993.17      -100.00    -1,225.75   -10,914.48
 202      -120.51    -1,005.24      -100.00    -1,225.75    -9,809.24
 203      -108.31    -1,017.44      -100.00    -1,225.75    -8,691.80
 204       -95.97    -1,029.78      -100.00    -1,225.75    -7,562.02
 205       -83.50    -1,042.25      -100.00    -1,225.75    -6,419.77
 206       -70.88    -1,054.87      -100.00    -1,225.75    -5,264.90
 207       -58.13    -1,067.62      -100.00    -1,225.75    -4,097.28
 208       -45.24    -1,080.51      -100.00    -1,225.75    -2,916.77
 209       -32.21    -1,093.54      -100.00    -1,225.75    -1,723.23
Summary for 2013:
  Interest  Paid: -1,216.19
  Principal Paid: -13,492.81
  Year Ending Balance: -1,723.23
  Sum of Interest Paid: -157,904.98
Pmt#     Interest    Principal       Prepay    Total Pmt      Balance
 210       -19.03    -1,106.72      -100.00    -1,225.75      -516.51
 211        -5.70      -516.51         0.00      -522.21         0.00

Total Interest: 157,929.71

</pre>

<p>In the second method, the periodic payment is increased by 100. With this method,
the annual summary table can also be computed.

<pre>
<>s
Financial Calculator
Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
Current Financial Calculator Status:
Compounding Frequency: (CF) 12
Payment     Frequency: (PF) 12
Compounding: Discrete (disc = TRUE)
Payments: End of Period (bep = FALSE)
Number of Payment Periods (n): 360              (Years: 30)
Nominal Annual Interest Rate (i): 13.25
  Effective Interest Rate Per Payment Period: 0.0110417
Present Value (pv): 100,000.00
Periodic Payment (pmt): -1,125.75
Future Value (fv): 0.00
Effective       Date: Thu Jun 06 00:00:00 1996(2450241)
Initial Payment Date: Thu Aug 01 00:00:00 1996(2450297)
<>pmt-=100
        -1,225.75
<>N
        210.42
<>
</pre>

<p>Thus, the loan will now be fully repaid in 210 full payments and a partial payment
as illustrated in the previous table.
To get the total interest paid, display the Annual Summary Amortization Schedule:

<pre>
Effective       Date: Thu Jun 06 00:00:00 1996
Initial Payment Date: Thu Aug 01 00:00:00 1996
The amortization options are:
The Old Present Value (pv)     was: 100,000.00
The Old Periodic Payment (pmt) was: -1,225.75
The Old Future  Value (fv)     was: 0.00
1: Amortize with Original Transaction Values
    and final payment: -1,742.55

The New Present Value (pve)  is:  100,919.30
The New Periodic Payment (pmt) is:  -1,237.02
2: Amortize with Original Periodic Payment
    and final payment: -10,967.39
3: Amortize with New Periodic Payment
    and final payment: -1,757.20
4: Amortize with Original Periodic Payment,
    new number of total payments (n): 218
    and final payment: -1,668.45

Enter choice 1, 2, 3 or 4: <>
</pre>

<p>Press '1' for option 1:

<pre>
 Amortization Schedule:
Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
Enter choice y, p or a:
<>
</pre>

<p>Press 'y' for an annual Summary

<pre>
Enter Filename for Amortization Schedule.
  (null string uses Standard Output):
</pre>

<p>Press enter to display the summary on the screen:

<pre>
  Amortization Table
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  Compounding Frequency per year: 12
  Payment     Frequency per year: 12
  Compounding: Discrete
  Payments: End of Period
  Payments (209): -1,225.75
  Final payment (# 210): -1,742.55
  Nominal Annual Interest Rate: 13.25
    Effective Interest Rate Per Payment Period: 0.0110417
  Present Value: 100,000.00
  Year      Interest   Ending Balance
  1996     -5,507.26       -99,378.51
  1997    -13,070.52       -97,740.03
  1998    -12,839.74       -95,870.77
  1999    -12,576.45       -93,738.22
  2000    -12,276.08       -91,305.30
  2001    -11,933.40       -88,529.70
  2002    -11,542.46       -85,363.16
  2003    -11,096.45       -81,750.61
  2004    -10,587.62       -77,629.23
  2005    -10,007.12       -72,927.35
  2006     -9,344.86       -67,563.21
  2007     -8,589.32       -61,443.53
  2008     -7,727.36       -54,461.89
  2009     -6,744.00       -46,496.89
  2010     -5,622.13       -37,410.02
  2011     -4,342.24       -27,043.26
  2012     -2,882.08       -15,216.34
  2013     -1,216.25        -1,723.59
  2014        -18.96             0.00

  Total Interest: -157,924.30
</pre>

<p>From the last line the Total interest has been decreased from $305,379.74 to
$157,924.30.

<p>We can also ask how much of a constant repayment would be necessary to  fully
repay the loan in 15 years and what would be the total interest paid?

<pre>
  <>n=12*15
          180
  <>opmt=pmt
          -1,125.75
  <>PMT
          -1,281.74
  <>pmt-opmt
          -155.99
</pre>

<p>Thus, a constant advanced repayment per periodic payment of $155.99 would fully
amortize the loan in 15 years.

<pre>
  <>a
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  The amortization options are:
  The Old Present Value (pv)     was: 100,000.00
  The Old Periodic Payment (pmt) was: -1,281.74
  The Old Future  Value (fv)     was: 0.00
  1: Amortize with Original Transaction Values
      and final payment: -1,279.73

  The New Present Value (pve)  is:  100,919.30
  The New Periodic Payment (pmt) is:  -1,293.52
  2: Amortize with Original Periodic Payment
      and final payment: -7,915.43
  3: Amortize with New Periodic Payment
      and final payment: -1,293.20
  4: Amortize with Original Periodic Payment,
      new number of total payments (n): 185
      and final payment: -1,738.05

  Enter choice 1, 2, 3 or 4: <>
</pre>

<p>Press '1' for option 1:

<pre>
   Amortization Schedule:
  Yearly, y, per Payment, p, or Advanced Payment, a, Amortization
  Enter choice y, p or a:
  <>
</pre>

<p>Press 'y' for an annual Summary

<pre>
  Amortization Table
  Effective       Date: Thu Jun 06 00:00:00 1996
  Initial Payment Date: Thu Aug 01 00:00:00 1996
  Compounding Frequency per year: 12
  Payment     Frequency per year: 12
  Compounding: Discrete
  Payments: End of Period
  Payments (179): -1,281.74
  Final payment (# 180): -1,279.73
  Nominal Annual Interest Rate: 13.25
    Effective Interest Rate Per Payment Period: 0.0110417
  Present Value: 100,000.00
  Year      Interest   Ending Balance
  1996     -5,501.01       -99,092.31
  1997    -12,987.86       -96,699.29
  1998    -12,650.80       -93,969.21
  1999    -12,266.27       -90,854.60
  2000    -11,827.58       -87,301.30
  2001    -11,327.09       -83,247.51
  2002    -10,756.12       -78,622.75
  2003    -10,104.72       -73,346.59
  2004     -9,361.57       -67,327.28
  2005     -8,513.75       -60,460.15
  2006     -7,546.51       -52,625.78
  2007     -6,443.04       -43,687.94
  2008     -5,184.14       -33,491.20
  2009     -3,747.93       -21,858.25
  2010     -2,109.42        -8,586.79
  2011       -383.38             0.00

  Total Interest: -130,711.19
</pre>

<p>The toral interest is reduced to $130,711.19. This compares to:

<ol>
<li>$130,711.19 - Fixed prepayment $155.99/period, 15 year term
<li>$152,964.13 - Variable Advanced Repayment, 15 year term
<li>$305,379.74 - no prepayment, 30 year term
</ol>

<a name="BalloonPmt">
<h2>Balloon Payment</h2></a>
<p>Balloon Payment
<p>On long term loans, small changes in the periodic payments can generate
large changes in the future value. If the monthly payment in the previous example is
rounded down to $1125, how much addtional (balloon) payment will be due
with the final regular payment.
<pre>
  <>s
  Financial Calculator
  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
  Current Financial Calculator Status:
  Compounding Frequency: (CF) 12
  Payment     Frequency: (PF) 12
  Compounding: Discrete (disc = TRUE)
  Payments: End of Period (bep = FALSE)
  Number of Payment Periods (n): 180              (Years: 15)
  Nominal Annual Interest Rate (i): 13.25
    Effective Interest Rate Per Payment Period: 0.0110417
  Present Value (pv): 100,000.00
  Periodic Payment (pmt): -1,281.74
  Future Value (fv): 0.00
  Effective       Date: Thu Jun 06 00:00:00 1996(2450241)
  Initial Payment Date: Thu Aug 01 00:00:00 1996(2450297)
  <>n=360
          360
  <>pmt=-1125
          -1,125
  <>FV
          -3,579.99
  <>
</pre>

<a name="CanadianMortg">
<h2>Canadian Mortgage</h2></a>
<p>Canadian Mortgage
<p>A "Canadian Mortgage" is defined with semi-annual compunding, <tt>CF == 2</tt>,
and monthly payments, <tt>PF == 12</tt>.

<p>Find the monthly end-of-period payment necessary to fully amortize a 25 year
$85,000 loan at 11% compounded semi-annually.
<pre>
 <>d
 <>CF=2
         2
 <>n=300
         300
 <>i=11
         11
 <>pv=85000
         85,000
 <>PMT
         -818.15
</pre>

<a name="EuropeanMortg">
<h2></h2></a>
<p>European Mortgage
<p>The "effective annual rate (EAR)" is used in some countries (especially
 in Europe) in lieu of the nominal rate commonly used in the United States
 and Canada. For a 30 year $90,000 mortgage at 14% (EAR), compute the monthly
 end-of-period payments. When using an EAR, the compounding frequency is
 set to 1.
<pre>
 <>d
 <>CF=1
         1
 <>n=30*12
         360
 <>i=14
         14
 <>pv=90000
         90,000
 <>PMT
         -1,007.88
</pre>

<a name="BiWeeklySav">
<h2>Bi-weekly Savings</h2></a>
<p>Bi-weekly Savings
<p>Compute the future value, fv, of bi-weekly savings of $100 for 3 years at a
 nominal annual rate of 5.5% compounded daily. (Set payment to
 beginning-of-period, bep = TRUE)
<pre>
 <>d
 <>bep=TRUE
         1
 <>CF=365
         365
 <>PF=26
         26
 <>n=3*26
         78
 <>i=5.5
         5.50
 <>pmt=-100
         -100
 <>FV
         8,489.32
</pre>

<a name="PV_AnnuiytDue">
<h2>Present Value - Annuity Due</h2></a>
<p>Present Value - Annuity Due
<p>What is the present value of $500 to be received at the beginning of each
 quarter over a 10 year period if money is being discounted at 10% nominal
 annual rate compounded monthly?
<pre>
 <>d
 <>bep=TRUE
         1
 <>PF=4
         4
 <>n=4*10
         40
 <>i=10
         10
 <>pmt=500
         500
 <>PV
         -12,822.64
</pre>

<a name="EffRate">
<h2>Effective Rate - 365/360 Basis</h2></a>
<p>Effective Rate - 365/360 Basis
<p>Compute the effective annual rate (%APR) for a nominal annual rate of 12%
 compounded on a 365/360 basis used by some Savings & Loan Associations.
<pre>
 <>d
 <>n=365
         365
 <>CF=365
         365
 <>PF=360
         360
 <>i=12
         12
 <>pv=-100
         -100
 <>FV
         112.94
 <>fv+pv
         12.94
</pre>

<a name="EffAPY">
<h2>Certificate of Deposit, Annual Percentage Yield</h2></a>
<p>Certificate of Deposit, Annual Percentage Yield
<p>Most, if not all banks have started stating return rates on Certificates of Deposit, CDs, as
an Annual Percentage Yoild, APY, and the nominal annual interest. For example, a bank will advertise
a CD with a 18 month term, an APY of 5.20% and a nominal rate of 5.00. What values of CF and PF will
are being used?

<pre>
 <>d
 <>n=365
         365
 <>CF=PF=365
         365
 <>i=5
         5
 <>pv=-100
         -100
 <>FV
         105.13
 <>CF=PF=360
         360
 <>fv+pv
         -5.20
</pre>
<a name="MortgPoints">
<h2>Mortgage with "Points"</h2></a>
<p>Mortgage with "Points"
<p>What is the true APR of a 30 year, $75,000 loan at a nominal rate of 13.25%
 compounded monthly, with monthly end-of-period payments, if 3 "points"
 are charged? The pv must be reduced by the dollar value of the points
 and/or any lenders fees to establish an effective pv. Because payments remain
 the same, the true APR will be higher than the nominal rate. Note, first
 compute the payments on the pv of the loan amount.
<pre>
  <>n=30*12
          360
  <>i=13.25
          13.25
  <>pv=75000
          75,000
  <>PMT
          -844.33
  <>pv-=pv*0.03
          72,750.00
  <>I
          13.69
  <>
</pre>

<a name="EquivPmt">
<h2>Equivalent Payments</h2></a>
<p>Equivalent Payments
<p>Find the equivalent monthly payment required to amortize a 20 year $40,000
loan at 10.5% nominal annual rate compounded monthly, with 10 annual
payments of $5029.71 remaining. Compute the pv of the remaining annual
payments, then change n, the number of periods, and the payment frequency,
PF, to a monthly basis and compute the equivalent monthly pmt.
<pre>
 <>d
 <>PF=1
         1
 <>n=10
         10
 <>i=10.5
         10.50
 <>pmt=-5029.71
         -5,029.71
 <>PV
         29,595.88
 <>PF=12
         12
 <>n=120
         120
 <>PMT
         -399.35
</pre>

<a name="Perpetuity">
<h2>Perpetuity - Continuous Compounding</h2></a>
<p>Perpetuity - Continuous Compounding
<p>If you can purchase a single payment annuity with an initial investment of
 $60,000 that will be invested at 15% nominal annual rate compounded
 continuously, what is the maximum monthly return you can receive without
 reducing the $60,000 principal? If the principal is not disturbed, the
 payments can go on indefinitely (a perpetuity). Note that the term,n, of
 a perpetuity is immaterial. It can be any non-zero value.
<pre>
 <>d
 <>disc=FALSE
         0
 <>n=12
         12
 <>CF=1
         1
 <>i=15
         15
 <>fv=60000
         60,000
 <>pv=-60000
         -60,000
 <>PMT
         754.71
</pre>

<a name="DevCo">
<h2>Investment Return</h2></a>
<p>Investment Return
<p>A development company is purchasing an investment property with an annual net cash
flow of $25,000.00. The expected holding period for the property is 10 years with an estimated
selling price of $850,000.00 at that time. If the company is to realize a 15% yield on the
investment, what is the maximum price they can pay for the property today?

<pre>
  Financial Calculator
  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
  <>CF=PF=1
          1
  <>n=10
          10
  <>i=15
          15
  <>pmt=25000
          25,000
  <>fv=850000
          850,000
  <>PV
          -335,576.22
</pre>

<p>So the maximum purchase price today would be $335,576.22 to achieve the desired yield.

<a name="Retiement">
<h2>Retirement Investment</h2></a>
<p>Retirement Investment
<p>You wish to retire in 20 years and wish to deposit a lump sum amount in an account
today which will grow to $100,000.00, earning 6.5% interest compounded semi-annually.
How much do you need to deposit?

<pre>
  Financial Calculator
  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
  <>CF=PF=2
          2
  <>n=2*20
          40
  <>i=6.5
          6.50
  <>fv=100000
          100,000
  <>PV
          -27,822.59
</pre>

<p>If you were to make semi-annual deposits of $600.00, how much would you need to deposit today?

<pre>
  <>pmt=-600
          -600
  <>PV
          -14,497.53
</pre>

<p>If you were to make monthly deposits of $100.00?

<pre>
  <>PF=12
          12
  <>n=20*12
          240
  <>pmt=-100
          -100
  <>PV
          -14,318.21
</pre>

<a name="PropVal">
<h2>Property Values</h2></a>
<p>Property Values
<p>Property values in an area you are considering moving to are declining at the rate
of 2.35% annually. What will property presently appraised at $155,500.00 be worth in 10 years
if the trend continues?

<pre>
  Financial Calculator
  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
  <>CF=PF=1
          1
  <>n=10
          10
  <>i=-2.35
          -2.35
  <>pv=155500
          155,500
  <>FV
          -122,589.39
</pre>

<a name="CollegeExpenses">
<h2>College Expenses</h2></a>
<p>College Expenses
<p>You and your spouse are planning for your child's college expenses. Your child
will be entering college in 15 years. You expect that college expenses at that time
will amount to $25,000.00 per year or about $2,100.00/month. If the child withdrew
the expenses from a bank account monthly paying 6% compounded on a daily basis (using
360 days/year), how much must you deposit in the account at the start of the  four
years?

<pre>
  Financial Calculator
  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
  <>CF=360
          360
  <>PF=12
          12
  <>n=12*4
          48
  <>i=6
          6
  <>pmt=2100
          2,100
  <>PV
          -89,393.32
</pre>

<p>Your next problem is how to accumulate the money by the time the child starts college.
You have a $50,000.00 paid-up insurance policy for your child that has a cash value
of $6,500.00. It is accumulating annual dividends of $1,200 earning 6.75% compounded monthly.
What will be the cash value of the policy in 15 years?

<pre>
  <>college_fund=-pv
          89,393.32
  <>d
  <>PF=1
          1
  <>n=20
          20
  <>i=6.75
          6.75
  <>pmt=1200
          1,200
  <>FV
          -48,995.19
  <>insurance=-fv+6500
          55,495.19
  <>college_fund-insurance
          33,898.13
</pre>

<p>The paid-up insurance cash value and dividends will provide $55,495.19 of the amount
necessary, leaving $33,898.13 to accumulate in savings. Making monthly payments into
a savings account paying 4.5% compounded daily, what level of monthly payments would be
needed?

<pre>
  Financial Calculator
  Copyright (C) 1990 - 1997 Terry D. Boldt, All Rights Reserved.
  <>d
  <>CF=360
          360
  <>n=PF*15
          180
  <>i=4.5
          4.50
  <>fv=college_fund - insurance
          33,898.13
  <>PMT
          -132.11
</pre>

<a name="refs">
<h1>References</h1></a>
<address>
PPC ROM User's Manual
<br>pages 148 - 164
</address>
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